You have found the following ages (in years) of 6 turtles. The turtles are randomly selected from the 33 turtles at your local zoo: $ 76,\enspace 35,\enspace 38,\enspace 71,\enspace 86,\enspace 66$ Based on your sample, what is the average age of the turtles? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we only have data for a small sample of the 33 turtles, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{196} + {729} + {576} + {81} + {576} + {16}} {{6 - 1}} $ $ {s^2} = \dfrac{{2174}}{{5}} = {434.8\text{ years}^2} $ We can estimate that the average turtle at the zoo is 62 years old. There is a variance of 434.8 years $^2$.